A counterexample to symmetry of $L^p$ norms of eigenfunctions
| Autor: | Beiner, Gabriel, Eagles, Nancy Mae, Verreault, William, Wang, Runyue |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We answer a question of Jakobson and Nadirashvili on the asymptotic behavior of the $L^p$ norms of positive and negative parts of eigenfunctions of the Laplacian. More precisely, we show that there exists a sequence of eigenfunctions $\psi_n$ on the flat $d$-torus for $d\geq 3$, with eigenvalues $\lambda_n\to\infty$ as $n\to\infty$, such that the ratio $\|\psi_n\chi_{\{\psi_n>0\}}\|_p / \|\psi_n\chi_{\{\psi_n<0\}}\|_p $ does not tend to $1$ as $n\to\infty$ for $1 |
| Databáze: | arXiv |
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